Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct

By Amaury Lambert. 2007

124 citations recorded by the source corpus.

We consider continuous-state branching (CB) processes which become extinct (i.e., hit 0) with positive probability. We characterize all the quasi-stationary distributions (QSD) for the CB-process as a stochastically monotone family indexed by a real number. We prove that the minimal element of this family is the so-called Yaglom quasi-stationary distribution, that is, the limit of one-dimensional marginals conditioned on being nonzero. Next, we consider the branching process conditioned on not being extinct in the distant future, or $Q$-process, defined by means of Doob $h$-transforms. We show that the $Q$-process is distributed as the initial CB-process with independent immigration, and that under the $L\log L$ condition, it has a limiting law which is the size-biased Yaglom distribution (of the CB-process). More generally, we prove that for a wide class of nonnegative Markov processes absorbed at 0 with probability 1, the Yaglom distribution is always stochastically dominated by the stationary probability of the $Q$-process, assuming that both exist. Finally, in the diffusion case and in the stable case, the $Q$-process solves a SDE with a drift term that can be seen as the instantaneous immigration.

Subfields: Artificial Intelligence, Management Science and Operations Research, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/464615a913eef39c20f9427d34e793a5992767fd

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Citations in the corpus

Old and New Examples of Scale Functions for Spectrally Negative Levy Processes (2007, background): “…in the theory of queuing and storage models Dube et al. (2004) and Bekker et al. (2007), in the theory of branching processes Bingham (1976) and Lambert (2007), in the theory of insurance risk and ruin Chiu and Yin (2005), Klüppelberg et al. (2004), Klüppelberg and Kyprianou (2006), Doney…”
Quasi-stationary distributions and population processes (2011, methodology): “In the general case where c 6= 0, it defines the so-called logistic Feller diffusion process following the terminology introduced by Etheridge [20] and Lambert [42].”
Population Dynamics and Random Genealogies (2008, background): “Before continuing further, we state a technical lemma, whose proof, as well as those of most other statements, can readily be found in Ref.[81].”
General criteria for the study of quasi-stationarity (2017, background): “However, these results do not apply to processes admitting several quasi-stationary distributions, which is known to happen in a variety of specific cases, even for processes irreducible in E (including branching processes [77, 2, 60, 63], one-dimensional birth and death processes [80, 37, 36, 85] a”
Quasi-stationary distributions and diffusion models in population dynamics (2007, background): “As in [3] (Theorem B), we can also describe the law of the process conditioned to be never extinct, usually called the Q-process (also see [19]).”
Special, conjugate and complete scale functions for spectrally negative Lévy processes (2007, background): “For example, in the theory of optimal stopping [1; 2] and [38], in the theory of optimal control [3], [40; 51] and [44], in the theory of queuing and storage models [21] and [4], in the theory of branching processes [10], [29] and [43], in the theory of insurance risk and ruin [15], [34], [33], [20]”
Stochastic equations of non-negative processes with jumps (2008, background): “(1.8) A particular case of this equation has been considered in Lambert (2007), where the uniqueness of the solution was left open.”
On the extinction of Continuous State Branching Processes with catastrophes (2013, background): “In other words, the process possesses positive jumps with infinite intensity [28].”